We show that, in terms of both implication and consistency strength, an extendible with a larger strong cardinal is stronger than an enhanced supercompact, which is itself stronger than a hypercompact, which is itself weaker than an extendible. All of these are easily seen to be stronger than a supercompact. We also study $C^{(n)}$ -supercompactness.

Bob Lubarsky and I have been working on this during the current academic year, and today we submitted it for publication. (Update: the paper has been accepted to the Archive for Mathematical Logic.) You can read a preprint here: ElementaryEpimorphisms.

Abstract

We show that every $\Pi_1$-elementary epimorphism between models of $ZF$ is an isomorphism. On the other hand, nonisomorphic $\Sigma_1$-elementary epimorphisms between models of $ZF$ can be constructed, as can fully elementary epimorphisms between models of $ZFC^-$. Elementary epimorphisms were introduced by Philipp Rothmaler.. A surjective homomorphism $f: M \to N$ between two model-theoretic structures is an elementary epimorphism if and only if every
formula with parameters satisfied by $N$ is satisfied in $M$ using a preimage of those parameters.

This is an adaptation of many of the results from the second chapter of my dissertation into a journal article. You can read the preprint here.

Abstract

I analyze the hierarchy of large cardinals between a supercompact cardinal and an almost-huge cardinal. Many of these cardinals are defined by modifying the definition of a high-jump cardinal. A high-jump cardinal is defined as the critical point of an elementary embedding $j: V \to M$ such that $M$ is closed under sequences of length equal to $\sup\{j(f)(\kappa) \ |\ f: \kappa \to \kappa \}$. Some of the other cardinals analyzed include the super-high-jump cardinals, almost-high-jump cardinals, Shelah-for-supercompactness cardinals, Woodin-for-supercompactness cardinals, Vopenka cardinals, hypercompact cardinals, and enhanced supercompact cardinals. I organize these cardinals in terms of consistency strength and implicational strength. I also analyze the superstrong cardinals, which are weaker than supercompact cardinals but are related to high-jump cardinals. Two of my most important results are as follows.

– Vopenka cardinals are the same as Woodin-for-supercompactness cardinals.

– There are no excessively hypercompact cardinals.

Furthermore, I prove some results relating high-jump cardinals to forcing, as well as analyzing Laver functions for super-high-jump cardinals.

UPDATE (2/25/15):

Kentaro Sato wrote to me recently and informed me that he has proved the result on the equivalence of Woodin-for-supercompactness cardinals and Vopenka cardinals in his paper,

What I call Woodin-for-supercompactness, he calls 0-W-huge in Def. A.1. In Cor. A.7, he shows this to be equivalent to what he calls 1-W-strong, which is also called 2-fold Woodin in Def. 9.1. In Cor. 10.6, he shows that 2-fold Woodin and 1-fold Vopenka are equivalent, with an even stronger statement, Thm. 10.5, the equivalence on the filter level.

I was not aware that Sato had proven this fact at the time that I wrote my paper.

My own modest contribution regarding the equivalence was to provide a different proof using a different theoretical framework, and to highlight the result using a different vocabulary.

This is my doctoral dissertation, submitted to the CUNY Graduate Center in April 2013. I plan to produce two journal papers summarizing and extending the material in the dissertation, one paper for each chapter. You can read it here.

Advisor: Joel David Hamkins

Abstract

This dissertation consists of two chapters, each of which investigates a topic in set theory, more specifically in the research area of forcing and large cardinals. The two chapters are independent of each other.

The first chapter analyzes the existence, structure, and preservation by forcing of inverse limits of inverse directed systems in the category of elementary embeddings and models of set theory. Although direct limits of directed systems in this category are pervasive in the set-theoretic literature, the inverse limits in this same category have seen less study. I have made progress towards characterizing the existence and structure of these inverse limits. Some of the most important results are as follows. An inverse limit exists if and only if a natural source exists. If the inverse limit exists, then it is given either by the entire thread class or by a rank-initial segment of the thread class. Given sufficient large cardinal hypotheses, it is consistent that there are systems with no inverse limit, systems with inverse limit given by the entire thread class, and systems with inverse limit given by a proper subset of the thread class. Inverse limits are preserved by forcing in both directions under fairly general assumptions but not in all cases. Prikry forcing and iterated Prikry forcing are important techniques for constructing some of the examples in this chapter.

The second chapter analyzes the hierarchy of the large cardinals between a supercompact cardinal and an almost-huge cardinal, including in particular high-jump cardinals. I organize the large cardinals in this region by consistency strength and implicational strength. I also prove some results relating high-jump cardinals to forcing. A high-jump cardinal is the critical point of an elementary embedding $j: V \to M$ such that $M$ is closed under sequences of length $\sup\{j(f)(\kappa) : \ \ \ f: \kappa \to \kappa\}$. Two of the most important results in the chapter are as follows. A Vopěnka cardinal is equivalent to a Woodin-for-supercompactness cardinal. There are no excessively hypercompact cardinals.

We present several generalizations of the well-known Kunen inconsistency that there is no nontrivial elementary embedding from the set-theoretic universe V to itself. For example, there is no elementary embedding from the universe V to a set-forcing extension V[G], or conversely from V[G] to V, or more generally from one ground model of the universe to another, or between any two models that are eventually stationary correct, or from V to HOD, or conversely from HOD to V, or indeed from any definable class to V, among many other possibilities we consider, including generic embeddings, definable embeddings and results not requiring the axiom of choice. We have aimed in this article for a unified presentation that weaves together some previously known unpublished or folklore results, several due to Woodin and others, along with our new contributions.